This section is intended to introduce various aspects of the art, which may be associated with exemplary embodiments of the present techniques. This discussion is believed to assist in providing a framework to facilitate a better understanding of particular aspects of the present techniques. Accordingly, it should be understood that this section should be read in this light, and not necessarily as admissions of prior art.
Hydrocarbons are widely used for fuels and chemical feed stocks. Hydrocarbons are generally found in subsurface rock formations that can be termed “reservoirs.” Removing hydrocarbons from the reservoirs depends on numerous physical properties of the rock formations, such as the permeability of the rock containing the hydrocarbons, the ability of the hydrocarbons to flow through the rock formations, and the proportion of hydrocarbons present, among others.
Often, mathematical models termed “simulation models” are used to simulate hydrocarbon reservoirs and optimize the production of the hydrocarbons. The goal of a simulation model is generally to simulate the flow patterns of the underlying geology in order to optimize the production of hydrocarbons from a set of wells and surface facilities.
The simulation model is a type of computational fluid dynamics simulation where a set of partial differential equations (PDEs) which govern multi-phase, multi-component fluid flow through porous media and the connected facility network is approximated and solved.
The set of governing differential equations vary with the physical process being modeled in the reservoir. They usually include volume balance, mass conservation, and energy conservation, among others. The simulation is an iterative, time-stepping process where a particular hydrocarbon production strategy is optimized.
Reservoir simulation models multiphase flows in reservoirs. Since geological models often are quite heterogeneous and may contain hundreds of millions of cells, upscaling is often used to reduce the model size. Reducing the model size facilitates simulation by reducing the size of the problem to be solved, thereby conserving computational resources. The upscaling step generates homogenized flow properties, for example, permeability, for computational cells.
When grid coarsening is done as a way to obtain fast simulation results for sensitivity analysis or history matching, the computational cells can be quite large and may also contain localized geological features, for example, shale barriers, faults, etc. The upscaled model, though easier to simulate, is only an approximation of the original reservoir model. The errors introduced in the upscaling process may be small when flow field is relatively smooth, but could be large otherwise. One challenge in upscaling is coarsening the reservoir model for fast computational performance while maintaining a high degree of accuracy.
There are different approaches to perform upscaling for reservoir simulation. One method is based on averaging (arithmetic averaging, harmonic averaging, etc.) permeability values inside the coarse cell. To improve accuracy, upscaling may be based on flows where permeability is obtained by solving local fine scale problems while assuming single phase, incompressible and steady-state flow, with either Dirichlet or Neumann or mixed boundary conditions. A common approach involves calculating coarse permeability or face transmissibility from flow rates obtained using fine scale pressure solutions and Darcy's law.
When upscaling permeability values, assumptions are typically made about pressure or flow profiles at the boundary. For example, constant pressure values of 1 or 0 may be assigned to a boundary perpendicular to the direction for which permeability or transmissibility is to be determined A no-flow condition may be assumed on the rest of the boundaries. Unfortunately, these assumptions may result in significant upscaling errors, which may become more severe as the simulation grids become coarser. To mitigate the problem, researchers have suggested techniques such as extended local upscaling, global upscaling, local-global upscaling, etc, which are more expensive computationally but improve the accuracy.
Recently, there have been a number of research articles published on the general subject of upscaling and multiscale methods. See, for example, Aarnes, J., “Multi-scale Simulation Of Flow In Heterogeneous Oil-Reservoirs,” Nov. 2004; see also, Crotti, M. A., Cobenas, R. H., “Upscaling of Relative Permeability Curves for Reservoir Simulation: An Extension to Area Simulations Based on Realistic Average Water Saturations,” Society of Petroleum Engineers, presented at SPE Latin American and Caribbean Petroleum Engineering Conference, Port-of-Spain, Trinidad, West Indies, 27-30Apr. 2003; see also Qi, D., Wong, P., Liu, K., “An Improved Global Upscaling Approach for Reservoir Simulation,” Petroleum Science & Technology, 2001, Volume 19, Issue 7pp. 779-795; Aarnes, J., Lie, K.-A., “Toward Reservoir Simulation on Geological Grid Models,” European Conference on the Mathematics of Oil Recovery, Cannes, France, 30Aug.-2Sep. 2004.
The following paragraphs of this Background section provide specific examples of known reservoir data analysis techniques. International Patent Application Publication No. WO2008008121 by ExxonMobil Upstream Research Co., discloses simulating a physical process, such as fluid flow in porous media, by performing a fine-grid calculation of the process in a medium. The fine grid solution is reused in subsequent coarse-grid calculations. For fluid flow in subsurface formations, the method may be used to optimize upscaled calculation grids formed from geologic models. The method decreases the cost of optimizing a grid to simulate a physical process that is mathematically described by the diffusion equation.
U.S. Pat. No. 6,823,297 to Jenny, et al., discloses a multi-scale finite-volume method for use in subsurface flow simulation. The method purports to solve elliptic problems with a plurality of spatial scales arising from single or multi-phase flows in porous media. Two sets of locally computed basis functions are employed. A first set of basis functions captures the small-scale heterogeneity of the underlying permeability field, and it is computed to construct the effective coarse-scale transmissibilities. A second set of basis functions is required to construct a conservative fine-scale velocity field.
The method efficiently captures the effects of small scales on a coarse grid, is conservative, and treats tensor permeabilities correctly. The underlying idea is to construct transmissibilities that capture the local properties of a differential operator. This leads to a multi-point discretization scheme for a finite-volume solution algorithm.
Transmissibilities for the MSFV method are preferably constructed only once as a preprocessing step and can be computed locally. Therefore, this step is well suited for massively parallel computers. Furthermore, a conservative fine-scale velocity field can be constructed from a coarse-scale pressure solution which also satisfies the proper mass balance on the fine scale.
A transport problem is ideally solved iteratively in two stages. In the first stage, a fine scale velocity field is obtained from solving a pressure equation. In the second stage, the transport problem is solved on the fine cells using the fine-scale velocity field. A solution may be computed on the coarse cells at an incremental time and properties, such as a mobility coefficient, may be generated for the fine cells at the incremental time. If a predetermined condition is not met for all fine cells inside a dual coarse control volume, then the dual and fine scale basis functions in that dual coarse control volume are reconstructed.
U.S. Pat. No. 6,826,520 to Khan, et al., discloses a method of upscaling permeability for unstructured grids. Permeabilities associated with a fine-scale grid of cells (representative of a porous medium) are scaled up to permeabilities associated with an unstructured coarse-scale grid of cells representative of the porous medium. An aerially unstructured, Voronoi, computational grid is generated using the coarse-scale grid as the genesis of the computational grid. The computational grid is then populated with permeabilities associated with the fine-scale grid.
Flow equations are developed for the computational grid, the flow equations are solved, and inter-node fluxes and pressure gradients are then computed for the computational grid. These inter-node fluxes and pressure gradients are used to calculate inter-node average fluxes and average pressure gradients associated with the coarse-scale grid. The inter-node average fluxes and average pressure gradients associated with the coarse grid are then used to calculate upscaled permeabilities associated with the coarse-scale grid.